# SphericalBesselJ

SphericalBesselJ[n,z]

gives the spherical Bessel function of the first kind .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• SphericalBesselJ is given in terms of ordinary Bessel functions by .
• SphericalBesselJ[n,z] has a branch cut discontinuity for noninteger in the complex plane running from to .
• Explicit symbolic forms for integer n can be obtained using FunctionExpand.
• For certain special arguments, SphericalBesselJ automatically evaluates to exact values.
• SphericalBesselJ can be evaluated to arbitrary numerical precision.
• SphericalBesselJ automatically threads over lists.

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

## Scope(37)

### Numerical Evaluation(4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

### Specific Values(4)

Limiting value at infinity:

SphericalBesselJ for symbolic n:

Find the first positive zero of SphericalBesselJ:

Different SphericalBesselJ types give different symbolic forms:

### Visualization(3)

Plot the SphericalBesselJ function for integer ( ) and half-integer ( ) orders:

Plot the real part of :

Plot the imaginary part of :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(12) is defined for all real and complex values: is defined for all real values greater than 0:

Complex domain is the whole plane except :

Approximate function range of :

For integer , is an even or odd function in depending on whether is even or odd:

This can be expressed as : is not an analytic function of for noninteger and negative values of :

SphericalBesselJ is neither non-decreasing nor non-increasing:

SphericalBesselJ is not injective:

SphericalBesselJ is neither non-negative nor non-positive: is singular for , possibly including , when is noninteger:

SphericalBesselJ is neither convex nor concave:

### Differentiation(3)

First derivative with respect to z:

Higher derivatives with respect to z

Plot the higher derivatives with respect to z:

Formula for the  derivative with respect to z:

### Integration(3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

### Series Expansions(6)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Find series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

### Function Identities and Simplifications(2)

Use FullSimplify to simplify spherical Bessel functions of the first kind:

Recurrence relations:

## Applications(1)

Solve the radial part of the Laplace operator in 3D:

## Properties & Relations(2)

SphericalBesselJ can be represented as a DifferentialRoot: